Rewrite as a simplified fraction. ${1.9\overline{91} = {?}}$
Answer: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 1991.9191...\\ 10x &= 19.9191...\end{align*}}$ Subtracting the two equations, we have: ${990x = 1972}$ ${x = \dfrac{1972}{990}} $ ${x = \dfrac{986}{495}} \text{or} {1 \dfrac{491}{495}}$